Two interpretations of the Lagrange Multiplier

Question

A question about the second answer on this thread: Help understanding Lagrangian multipliers?

If we have a standard utility maximizing problem $$ \max_{x,y} U(x,y) $$ with the constraint $p_{x}x + p_{y}y = B$, where $p_{x}, p_{y}$ are the prices of $x,y$, and $B$ is the budget/income. The Lagriangian would work as $$ U(x,y) + \lambda(B-p_{x}x - p_{y}y) $$ and we would proceed to take partial derivatives and find critical points. My question is on the interpretation of the Lagrangian function above and specifically the multiplier $\lambda$:

1) In what way is $\lambda$ the "price we pay for not obeying our constraint"? (as one of the explanations in the linked post has it)

2) Intuitively, not mathematically, how is this equivalent to the interpretation of $\lambda$ as a "shadow price"? What exactly is a shadow price?

3) What difference does it make to $\lambda$ if our constraint was written as $p_{x}x + p_{y}y - B$ instead? (i.e. in reverse order)

Answer 1

If you would NOT obey the constraint and spend 1\$ more (or more precisely an infinitesimal amount more) your utility would go up by $\lambda$. As such $\lambda$ is what you give up for obeying the constraint, it is the price you pay indirectly and hence it is called the shadow price.

Writing the constraint in reverse has no effect other than changing the sign of $\lambda$, but that depends on the way you set-up your problem anyway (I.e. whether you add the constraint with a plus or a minus).